Journal of Applied Mathematics and Physics, 2014, 2, 61-71
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
How to cite this paper: Lipovka, A. (2014) Planck Constant as Adiabatic Invariant Characterized by Hubble’s and Cosmolog-
ical Constants. Journal of Applied Mathematics and Physics, 2, 61-71. http://dx.doi.org/10.4236/jamp.2014.25009
Planck Constant as Adiabatic Invariant
Characterized by Hubble’s and
Department of Investigation for Physics, Sonora State University, Hermosillo, Sonora, Mexico
Received January 2014
Within the framework of Einstein-Cart an -Shrödinger formalism with asymmetric connections, the
Planck constant is calculated from the first principles (from geometry of our Universe), as the
adiabatic invariant of electromagnetic field on the Riemann-Cartan manifold. The Planck constant,
calculated with actually measured cosmological parameters, coincide with that one, measured in
laboratory with precision up to the second digit. The non-local generalization of quantum theory
is suggested. The fundamental sense of the Quantum Theory is discussed, and physical sense of the
cosmological constant is revealed. Within the mentioned framework, the quantum theory is natu-
rally unified with gravity.
Cosmology, Quantum Theory, Unified Th eory
Quantum Theory (In accordance with the historical terminology, we shall call “Quantum Theory” (QT) the
theory, based on the concept of wave functions, or probability amplitudes), that recently has celebrated its
100-year anniversary, allowed at the time to overcome a crisis that happened in atomic physics, giving research-
ers a necessary tool for the calculation of atomic and subatomic phenomena with an accuracy which is in strik-
ing agreement with experiment. However, since its foundation and up to now, physicists and mathematicians are
still attempting to interpret this unusual QT formalism. On the one hand Quantum Mechanics (QM) from the
beginning (and then Quantum Field Theory as its successor) was built on the axiomatic approach, which cannot
be considered as satisfactory. So, the concept of the wave function was postulated for all describable entities. On
the other hand, the evolution operator for a system is linear with the wave function, whereas its square appears
as the result of the measurement process. If we add to the above the presence of divergences and unrenormaliza-
bility of theories in general case, the serious problems with unification of QT and general relativity (GR), and
inability to obtain the mass and charge from the first principles, the incompleteness of QT becomes apparent,
and thus there is need to find a complete theory describing the atomic and nuclear systems.
Since the moment of discovery, QT did not please its creators, giving rise to numerous discussions about the
place for probability in physics, the wave-particle duality, discussion of thought experiments and paradoxes. We
shall not discuss here again the well-known history of QT, for that the reader should refer to the monograph .
With such an “unu sua l” physics, researchers put up nearly a century, excusing its numerous defects, because QT
allows calculate physically interesting phenomena in excellent agreement with the experiment. The situation
began to change in the last decade of the 20th century, when the crisis that hit theoretical physics became ob-
vious to many physicists and people started talking loudly about the problems that arise when trying to unify QT
and GR. Among the most serious problems of the Standard Model we could mention the following:
• The problem of the collapse of the wave function (the problem of the observer, or Einstein-Podolsky-Rosen
• The presence of unrenormalizable (in general) dive r genc e s.
• The huge discrepancy between the calculated with QFT methods and observed cosmological constant.
• Conflict of QT with GR at the horizon of black holes.
• Recent experimental data obtained with the Planck satellite, which disfavors all the best-motivated inflatio-
nary scenarios .
• Inability of a reasonable harmonization or unification of the standard model with gravity.
This incomplete list of problems indicates very serious gaps in our understanding of Nature. For the most
cases, the problems are directly or implicitly appear from a misunderstanding of the basis of the quantum theory,
and the nature of its main concepts and axioms. The present paper is urged to fill the above mentioned gap and
to specify a way free from the difficulties listed above. We begin with a generalization of the quantum theory
because in its present form it cannot be unified with General relativity. In second part of the paper (Sections 5, 6
and 7) we show how the QT and relativity should be unified and calculate the Planck constant (fundamental pa-
rameter of QT) from the geometry of our Universe, i.e. from the Hubble and cosmological constants.
It is well known that quantum mechanics appeared from the need to explain the experimentally observed black-
body emission spectrum and atomic spectra. Planck was the first who propose an analytical formula to describe
the spectral energy distribution which was consistent with the experiment. However, as it was noted by Einstein
, the way in which Planck obtained his result, was not quite correct, though it led to the correct result. The
problem was that Planck included in his formula not only the electromagnetic field, but also oscillators associated
with the matter. As a result, in the electrodynamic part, based on Maxwell’s equations, the energy of the oscilla-
tors was a continuously varying value, while in the statistical part the same energy was considered as a discrete
value (qua nt i zed ).
In 1905 Einstein published the work  in which he showed that the emission field (without any assumptions
on the nature of matter) behaves so as if consists of separate quanta (photons), characterized by energy hv. Later,
in 1910 Debye  argued that Planck’s formula can be deduced for the pure radiation field, absolutely without
any assumptions on the oscillator’s properties of the substance. Thus Planck’s law and all its consequences, fol-
lows from the fact that the energy of freely propagating electromagnetic field is divided by parts proportional to
hv. Recently this result was confirmed experimentally . This fact is the only we need to obtain the Planck
constant from the geometry of our Universe, so reader interested in the relation between the Planck constant and
local curvature of our Universe (Cosmological constant and Hubble constant) is referred directly to the chapter
“Adiabatic invariant” of present paper. But now we consider consequences for the QT, which follows from the
mentioned above properties of the electromagnetic field.
It is known that the Bohr-Sommerfeld theory (so-called old quantum theory), based on the adiabatic hypothe-
sis, is founded on two quantum axioms, which when added to the axioms of classical mechanics allow us to
build a quantum theory. These two axioms are written as:
p dqn h=
The hypothesis expressed by Sommerfeld (that action for any elemental process is changed by value h) served
as the basis to write these relations. It states that in each elementary process, the action of the atom changes by
an amount equal to the Planck constant. However, if we take into account the results obtained by Einstein and
Debye, we easily receive these postulates, as a consequence of classical mechanics, i.e. we can construct the
reasonable classical theory of emission/absorption in lines, and the classical atomic theory without recurring to
the concept (axiom) of the wave function and the problems provoked by last one. It should be stressed here, that
so-called “new quantum theory” also is based on the axiom, and this axiom (of the wave functions existence)
cannot be explained or reduced to real physics, whereas the Bohr-Sommerfeld axioms can be reduced to (or ob-
tained from) classical physics, which provide us with a fundamental view to the basic concept and understanding
the nature of QT. To achieve the above, it should be noted, that there are only two fields which are carrying out
interactions at distances we are interested in
. These are the electromagnetic and the gravitational
fields. Considering that the interaction constant for a gravitational field is negligible in comparison with the
electromagnetic one, we can surely approve the following: Everything that we see, feel, hear, measure, register
with detectors, is the electromagnetic field and nothing else. That is we perceive the real world in the form of
this picture, by means of electromagnetic waves registered by us. It is important to understand, that the electro-
magnetic field acts as intermediary between the observer and the real (micro ) world, hiding from us reality (idea
of existence of the so-called hidden variables or beables in QM). In our case these hidden variables lose the
mystical meaning, becoming usual classical variables—coordinates and momenta of particles, but which can be
measured only by the electromagnetic field means.
Thus as a starting point we propose the following:
• The electromagnetic field is the only field responsible for interaction between objects and observer in Quan-
• The electromagnetic field is quantized without the need of any assumptions about the properties of oscilla-
tors. That is the Planck’s relation of
is satisfied, irrespective of the oscillators properties
The last thesis means that there exists (and therefore can be emitted) only the photon possessing the period
. In other words, emission/absorption of a photon can occur only for the whole period of movement of a
charge (in system of coordinates in which proceed the emission/absorption).
Let ’s consider the closed system in which charge moves harmonically and with constant acceleration. In this
case the Hamilton function of the electron does not depend explicitly on time. Let’s write it down as:
here K, U are kinetic and potential energy and E is a total energy of system. Then function of Lagrange is:
2L KUKE=−= −
Let ’s write down classical action for the bounded electron:
are the periods of movement of the electron in system on the first and second orbit respec-
tively. Then, considering the equation of Hamilton-Jacobi, for two different orbits 1 and 2 we have
21 21 2 211
However (see statements 1 and 2, mentioned above)
ETE ThvTh−= =
is action for a emitted/absorbed photon. Thus
that actually represents the first axiom of Bohr-Sommerfeld (1).
Let us consider for example an electron in the central field in the nonrelativistic limit. We have:
. Then expression (7) gives
which for s-state of atom of hydrogen gives a known ratio
Or, the same
are the angular momenta. To write down the last expression we used the fact that the ob-
formally coincides with the angular momenta for the electron in the coulomb field.
Let ’s put
(that corresponds to
). In this case we have
, so we
MM MM=+== +=
From this expression and a principle of mechanical similarity for the central potentials of
, we have
from where it follows:
Then for a classical harmonic oscillator
from (11) we get:
and for atom of hydrogen
The value E₁ in the last expression can be found easily from expression (6)
Accepting classical value of the period
and taking into account (13)
Thus we showed that so-called quantization of system (axioms of Bohr-Sommerfeld) arises in absolutely
classical way from the intrinsic properties of the electromagnetic field and cannot be treated as quantum proper-
ty of space or matter.
3. Harmonic Oscillator
There is a common misconception that the addition term of 1/2, which appears in the energy of the harmonic os-
cillator, is a quantum effect and is associated with the so-called zero-oscillations. Due to the methodological
importance of this question, we discuss it here in a little more detail in the non-relativistic limit, and show that it
is a purely classical effect.
Accordingly to classical mechanics, the energy of the harmonic oscillator is:
Considering that for the harmonic oscillator average value
, we obtain for the average energy for the
To carry out transition from an initial state of system to the final one
, we should “take away”
energy from our oscillator by electromagnetic field.
It is known that emission of an electromagnetic field by a moving charge differ from zero only if we integrate
for the full period T of movement in the course of which the emission or absorption appears. It corresponds to
the fact that the full photon instead of a part is emitted/absorbed, that is the generated field satisfying to a peri-
The factor of proportionality between energy and frequency for a free electromagnetic field is ℏ:
∆= − =
(Once again we emphasize here that as it follows from Einstein’s and Debye works, the constant ℏ concerns
only to the electromagnetic field and do not appears in any way from matter properties, or the size of the system
under consideration). Expression (12) gives a ratio between energy levels, however from Equation (18) it is
clear that the residual energy
cannot be emitted by a photon with energy
2 2E EEmrmrE
Therefore this additive constant (which appears owing to the shape of the potential) should be simply added to
the expression (12):
Thus, the additive constant 1/2 appears naturally from classical consideration.
4. Quantum Mechanics Is the Fourier-Transformed Classical Mechanics
In standard textbooks of quantum mechanics problems arise and are solved for isolated systems, when the
free/bounded electromagnetic field is not included in the Hamiltonian of the system. For example a harmonic
oscillator, the hydrogen atom, molecular potentials, etc. Thus on the one hand any changes in the system (transi-
tions between levels) are associated with the photons, but on the other hand, this electromagnetic field does not
appear in such Hamiltonian. Reasonable questions arise: where the electromagnetic field is and why it does not
appears in the Hamiltonian H? How these electromagnetic fields are taken into account for the quantization of
such a systems?
At the beginning of the 20-th century the equations describing the quantum system have been intuitively
guessed and accepted for the calculations (despite the emerging issues). It became possible because their pre-
dicted results were perfectly consistent with the experiments at the time. However, the meaning of the wave eq-
uations and the wave function itself is still not completely understood. In this section we will show sense of the
formalism of quantum mechanics making a start from bases of classical mechanics
For simplicity, consider the one-dimensional motion. The generalization to three dimensions is obvious. Sup-
pose we have the classical equation for energy of system:
Here H—is its classical Hamilton function and E—is the total energy of the system. Let’s consider a particle
in the field of U(x). For a total energy of system we have two possibilities:
1) E < 0 the system is bounded, we have a periodic movement,
2) E > 0 the system is unbounded, we have a free movement.
Any function (and the Hamilton one in particular) can be expanded in a Fourier series (E < 0) or integral (E >
0) in the complete set of functions. Photons in turn can be described by harmonic waves which form such com-
plete set of functions for the expansion we interested in:
exp ik x
are 4 -vectors.
Consider E < 0, that corresponds to a discrete spectrum in quantum mechanics. The case of continuous spec-
trum, when E > 0, differs only by replacement of sums by integrals, but the entire derivation of the equations is
Let ’s apply to Equation (21) the Fourier-transform on coordinate
( )()( )
, ,d,dHkxkx xEkx x
2 ed eded
px Etpx Etpx Et
from where we obtain:
is the Hamilton operator of the system under consideration.
We note here that the replacement of an electron by a positron (formally changes the sign in the exponent for
opposite one), leads to the replacement of t by
in Equation (25). In Equation (26) in the brackets we have
the Hamilton operator
, which actually is the Liouville operator (it corresponds to electron in potential U(x),
without the coupled electromagnetic field), so it has a complete set of orthogonal eigenfunctions.
be a complete set of eigenfunctions of the operator
, then we can write down
and the Equation (26) becomes
or in non-relativistic limit one can write:
( )( )
This is the equation of Schrödinger in coordinate representation. It is clear that if in Equation (23) we inte-
grate on p instead of coordinate, in the same way we obtain the Schrödinger equation, but now in p-representa-
( )( )
Let ’s make now inverse transformation of expression (28). We have:
we can obtain
or in another form:
That immediately implies matrix notation of quantum mechanics.
So, we have shown that:
• The quantum mechanics is the Fourier-transformed classical mechanics, and transformation goes on the
function of the electromagnetic field which does not appear obviously in the Schrödinger equations, remain-
ing out of consideration framework.
• The quantum theory is an incomplete (local) theory because it is based on an incomplete (local) Equation (29)
(of Schrödinger) instead of the complete (non-local) Equation (28) where the electromagnetic field appears
as expansion coefficients
under summation and integration.
• So-called wave functions are not “probability density” but are just eigenfunctions of the operator of Liou-
ville which forms the problem of Sturm-Liouville for incomplete system. And these are the eigenfunctions
that allow us to make decomposition of the bounded electromagnetic field coupled with a charge to include
it into consideration. It should be stressed here, the theory based on the Equation (28) does not suffer of the
wave function collapse problem, and the Einstein-Podolsky-Rosen paradox does not appears. In consequence
with the expression (28), the wave function is defined completely by the photon, and its collapse appears
within the wavelength size of the photon (integration on dx in the expression (28)). So, within this complete
theory there do not appear movements characterized by velocities more than light speed, as it take place in
the Einstein-Podolsky-Rosen paradox for the Schrödinger Equation (29).
To conclude, the uncertainty principle
should be mentioned briefly. As it was discussed above,
any measurement occurs with the assistance of a photon. In this way, we can measure the coordinates of the ob-
ject with the precision of up to
where λ is wavelength of the photon. However in the course of
coordinate measurement the photon transfers a part of their impulse to the measured object so we can write
. Combining the first expression with the second one we have
. On the other hand, in
view of that the phase is an invariant, we can conclude that symmetric expression also take place
5. Adiabatic Invariant
From astronomical observations it is well established that we live in a non-stationary Universe, in which all pa-
rameters change over time. By taking into account this fact, let’s consider an isolated mechanical system making
finite movement. Without loss of a generality we consider only one coordinate q, characterizing movement of
the system. Suppose also that movement of the system is characterized by a certain parameter r. Here we can
—radius of the Universe or
—scalar curvature of space. The final result will not depend on
Let the parameter r adiabatically change with time, i.e.
where T—is the characteristic time, or period of motion of our system. From this relation one can obtain estima-
tion for the proper frequency of the system satisfying the adiabatic condition:
which actually corresponds to the always fulfilled relation
(the wavelength of a photon is much less
than the size of the Unive r se ). It is clear that the system in question (photon) in this case is not isolated, and for
the total system energy we have the linear relationship
. The Hamiltonian of the system, in turn, depends
on parameter r, therefore
Averaging this expression on the period, we obtain
or designating our adiabatic invariant by h, get from this expression
whe r e
is the Planck’s constant on their sense. Considering that
we can write down the energy of a photon as
It should be noted here, that the integration constant
for general case.
6. Relation between the Geometry of the Universe and the Value of Planck
Earlier we have shown how the quantum mechanical picture of surrounding reality appears. In the present sec-
tion we obtain the important quantitative characteristic of the quantum theory—value of the Planck constant,
from observable geometry of the Universe.
It is well known that General Relativity formulated on Riemann manifold has some difficulties. Among the
most significant the following should be mentioned:
• The presence of singularities.
• Inability to take into account the “large numbers” of Eddington-Dirac which formally suggest a relation be-
tween cosmological and the quantum values.
• The cosmological constant which has no explanation within the framework of GR.
To search for a solution of these problems we must consider more general extensions of the Riemann geome-
try. One of its possible natural extensions is the geometry of Riemann-Cartan in which the theory of Einstein-
Cartan with asymmetrical connections can be developed. There is a variety of reasons for such a choice:
1) The theory of Einstein-Car tan satisfies the principle of relativity and also the equivalence principle and
does not contradict the observational data.
2) It follows necessarily from gauge theory of gravitation.
3) It is free from the problem of singularities.
4) It suggests the most natural way to explain the cosmological constant as a non-Riemannian part of the sca-
lar curvature of space, caused by torsion.
Within Riemann’s geometry, as it is known, for the tensor of electromagnetic field we have relation:
ν µµνν µµν
(Due to the symmetry of connections, the covariant derivatives of 4-potencial in the field tensor can be subs-
tituted by partial derivatives). But in the case of Einste in-Ca r tan theory with asymmetrical connections, the rela-
tion (42) is not more fulfilled and an additional term in the tensor of electromagnetic field appears.
To construct a theory we need the Lagrangian, which includes a natural linear invariant—the scalar curva-
ture obtained by reduction of the Riemann-Cartan tensor of curvature. Let’s accept from the beginning that
curvature of space is small (that conforms to experiment) and, therefore, we can neglect by quadratic inva-
riants in Lagrangian, having written down action for a gravitational field and a matter in Riemann-Cartan
geometry this manner:
Here c—light velocity, G—gravitational constant, g—determinant of the metric tensor
is the Lagrangian of the matter which have been written down for Riemann-Cartan manifold,
. Varying it we obtain
is a tensor of density of energy-mo mentum of the matter in geometry of R-C. Simplifying on in-
dexes we have:
or in other form
where R—is the scalar formed of the Riemann’s tensor,
—trace of tensor
tromagnetic field in R-C geometry.
In the right side of Equation (47) we have the value associated with the difference of geometry from the Rie-
mann one (the trace of a tensor
for the electromagnetic field is equal to zero in Riemann geometry because
of symmetry of connections) that we want to evaluate. The problem of the direct estimation of the value of
is that we do not know the true metric of the Universe we live in. We also do not know the real connection coef-
ficients of our space. For this reason, we cannot directly calculate the value that we are interested in. Accor-
dingly, we cannot just write out a corresponding amendment to the energy of electromagnetic field. However we
can estimate this value indirectly, considering that the left part of expression (47) contains observable values.
As it follows from the section “adiabatic invariant” for the action of electromagnetic field we have:
SS h= −
is an constant of integration and h—is the adiabatic invariant (Planck constant) caused by slowly
changing curvature of space in the Riemann-Cartan Universe. Then, considering that the trace of the tensor
for the electromagnetic field is equal to zero in Riemann’s geometry, we can write at once from (47)
We emphasize here that on the left side of this expression, we have the observed quantities which characterize
the Universe geometry, while on the right side, appears the Planck constant, which in turn, characterize a mi-
crocosm. The value
is minimum possible interval of time corresponding to action h. To find it we notice
that energy of the corresponding electromagnetic field can change only by the value hν. (see first part of the pa-
per). Let’s consider as an example the atom of hydrogen (for our purposes we could consider any system). The
first Bohr orbit is characterized by value
is the electron mass,
—velocity of the electron at first Bohr orbit. State with
is not achievable for our system. As
radius reduces from
to the Compton wavelength
, the value
cannot be changed, for the
photon cannot be emitted. So we can write
we need to emphasize especially: time, as well as space, are continuous, i.e. they do not quantized. The interval
is the minimum interval of time, corresponding to value h. From expression (49), we
whe r e
Let ’s estimate the Planck constant. The measured values of the Hubble constant were presented in paper 
74.2 3.6Hkms Mpc
and paper 
73.8 2.4Hkms Mpc
. Let’s take for our assessment av-
. Cosmological constant
we adopt according to measurements
and we accept critical density
. Then, from expression (50) we obtain value for the
6.6 10herg s
, that coincides to within the second sign with experimental value.
Recently, the issue of a possible change of the fine structure constant α on time is widely debated, so for con-
venience, we put here another interesting relationship, which follows from (50)
7. Other Observational Effects
The results suggested in present work can be proved by independent experiments. The most basic of them is of
course the double slit experiment. Recently it was accurately carried out by , which clearly argued for our
model of non-local quantum theory. Another possible experiment could be a measurement of the blackbody
spectrum in far Reyleigh-Jeans region. As it was shown earlier, if the geometry of Riemann-Cartan has non-zero
scalar curvature, in expression for energy of electromagnetic field appears the additional term hν. The energy of
one photon in this case is:
where ν is a frequency of a photon, and a small additional energy
Integration here is carried out over the volume of one photon. Intensity of the black body emission in this case
one can write as
As one can see, in Wien and in close Reyleigh-Jeans regions the spectrum is almost coincide with Planck one
because of small value of
. However it is clear that the small additive energy
can lead to some devia-
tions from Planck spectrum in far Reyleigh-Jeans region and, probably, such deviation could be measured expe-
rimentally. It is necessary to emphasize that such experiment has independent great importance because will al-
low to state an assessment to the value
and to throw light on the geometrical nature of electromagnetic
In present work we made the next logical step towards implementation of the program started by Einstein and
Schrödinger in the fifties of the XX century (model of Einstein-Cartan-Schrödinger). Namely we show that the
Planck constant is actually the adiabatic invariant of the electromagnetic field, characterized by scalar curvature
of space of the Riemann-Cartan geometry. The main results of present work are:
For the first time we obtained the ratio between Riemannian scalar curvature of the Universe, the Cosmologi-
cal constant and Planck’s constant (see expression (50)), true up to the second decimal place (
It is stressed that due to change of geometry of the Universe, the Planck constant changes with time too.
The physical sense of the cosmological constant, as the no-Rie mannian part of the scalar curvature, which
appears due to the presence of torsion (asymmetrical connections), is revealed for the first time.
Dependence of the fine structure constant on the total scalar curvature of the Universe is obtained (52).
Within the used framework, natural unification of gravitation with the quantum theory is obtained.
The spectral density of the blackbody radiation is obtained in linear on curvature invariants approach.
Bases of the quantum theory are reconsidered and the physical sense of wave function is found. It is shown
that if we eliminate an unnatural axiom of existence of wave function of matter, the huge discrepancy between
calculated by the QFT methods and observed cosmological constant, disappears.
The approach based on the Equation (28), completely removes a problem of collapse of wave function and
classically resolves the Einstein-Podolsky-Rosen paradox. According to expression (28) “wave function”, as it
should be, is completely determined by corresponding electromagnetic field and its collapse occurs at scales of
wavelength of the photon (integration on dx in expression (28)). Thus in this way there is no need for transmis-
sion of a signal with a superluminal speed as it takes place in paradox of E-P-R for the Schrödinger equation
I would like to acknowledge Dr. J. Saucedo for the valuable criticism and comments. I also grateful to the Pul-
kovo Observatory team and particularly to Dr. E. Poliakow for the opportunity to spent part of 2008 year in the
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